Most extant localization theories for spaces, spectra and diagrams
of such can be derived from a simple list of axioms which are verified
in broad generality. Several new theories are introduced, including
localizations for simplicial presheaves and presheaves of spectra at
homology theories represented by presheaves of spectra, and a theory
of localization along a geometric topos morphism. The
$f$-localization concept has an analog for simplicial presheaves, and
specializes to the $\hbox{\Bbbvii A}^1$-local theory of
Morel-Voevodsky. This theory answers a question of Soul\'e concerning
integral homology localizations for diagrams of spaces.

In this paper, we define the notion of $R_{\ast}$-$\LS$ category
associated to an increasing system of subrings of $\Q$ and we relate
it to the usual $\LS$-category. We also relate it to the invariant
introduced by F\'elix and Lemaire in tame homotopy theory, in which
case we give a description in terms of Lie algebras and of cocommutative
coalgebras, extending results of Lemaire-Sigrist and F\'elix-Halperin.

The notion of shape fibration with the near lifting of near
multivalued paths property is studied. The relation of these
maps---which agree with shape fibrations having totally disconnected
fibers---with Hurewicz fibrations with the unique path lifting
property is completely settled. Some results concerning homotopy and
shape groups are presented for shape fibrations with the near lifting
of near multivalued paths property. It is shown that for this class of
shape fibrations the existence of liftings of a fine multivalued map,
is equivalent to an algebraic problem relative to the homotopy, shape
or strong shape groups associated.

In the first paper with the same title the authors
were able to determine all partially oriented flag
manifolds that are stably parallelizable or
parallelizable, apart from four infinite families
that were undecided. Here, using more delicate
techniques (mainly K-theory), we settle these
previously undecided families and show that none of
the manifolds in them is stably parallelizable,
apart from one 30-dimensional manifold which still
remains undecided.

Let $M(X,Y)$ denote the space of all continous functions
between $X$ and $Y$ and $M_f(X,Y)$ the path component
corresponding to a given map $f: X\rightarrow Y.$ When $X$ and
$Y$ are classical flag manifolds, we prove the components of
$M(X,Y)$ corresponding to ``simple'' maps $f$ are classified
up to rational homotopy type by the dimension of the kernel of
$f$ in degree two
cohomology. In fact, these components are themselves all products
of flag manifolds and odd spheres.